The Mathematics of Music

The musical notes between one octave on the next are set up so that the ratio between the frequency of one note and the frequency of the next higher note is the same. Let's call this ratio \(r \) and so we have, starting with the notes \(G_1, Ab, A \):$$ \frac{f_{Ab}}{f_{G_1}} = r \text{ and } \frac{f_A}{f_{Ab}}=r \text{  and so   } f_A=r^{\scriptscriptstyle{2}} \times f_{G_1} \text{ etc.} $$In the end, we'll have the following crucial relationship between one octave and the next:$$ f_{G_2}=r^{\scriptscriptstyle{12}} \times f_{G_1} \text{ but because }f_{G_2}=2 \times f_{G_1} \text{ we have } r^{\scriptscriptstyle{12}}=2 \text{  or } r=\sqrt[12]{2}$$The perfect fifth, which according to Pythagoras should be exactly halfway between the two octaves (or seven semitones) giving a frequency of: $$ \frac{\scriptstyle{3}}{\scriptstyle{2}} \times f_{G_1} \text{ compared to the actual } 2^{\scriptscriptstyle{7/12}} \times f_{G_1} \approx 1.498307077 \times f_{G_1}$$Thus the two are almost identical but of course Pythagoras applied his 1.5 method to determine all the other notes but this is not the method that the equal temperament scale uses.